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This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

Given a collection of hypergraphs H=(H1,...,Hm)${{\bf H}}=(H_1, \ldots , H_m)$ with the same vertex set, an m$m$‐edge graph F⊂∪i∈[m]Hi$F\subset \cup _{i\in [m]}H_i$ is a transversal if there is a bijection ϕ:E(F)→[m]$\phi :E(F)\rightarrow [m]$ such that e∈E(Hϕ(e))$e\in E(H_{\phi (e)})$ for each e∈E(F)$e\in E(F)$. How large does the minimum degree of each Hi$H_i$ need to be so that H${{\bf H}}$ necessarily contains a copy of F$F$ that is a transversal? Each Hi$H_i$ in the collection could be the same hypergraph, hence the minimum degree of each Hi$H_i$ needs to be large enough to ensure that F⊆Hi$F\subseteq H_i$. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac‐type results for (powers of) Hamilton cycles. For example, we derive that any collection of rn$rn$ graphs on an n$n$‐vertex set, each with minimum degree at least (r/(r+1)+o(1))n$(r/(r+1)+o(1))n$, contains a transversal copy of the r$r$th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture. [ABSTRACT FROM AUTHOR]